Additive Identity
ArithmeticThe additive identity is zero: adding zero to any number leaves that number unchanged.
Formula
a + 0 = a
Definition
The additive identity is zero. When you add zero to any number, the number stays the same.
Example
$7 + 0 = 7$. $0 + 100 = 100$. No matter what number you start with, adding zero never changes it.
Key Insight
Zero is the "do-nothing" number for addition. It is the identity because the result is identical to what you started with.
Definition
The additive identity of a number system is the element $e$ such that $a + e = e + a = a$ for all $a$ in the system. In the integers, rationals, reals, and complex numbers, the additive identity is $0$. The additive identity is unique in any group.
Example
In $\mathbb{Z}/5\mathbb{Z}$: $3 + 0 = 3$, $0 + 3 = 3$. Zero is still the additive identity. In a vector space, the zero vector is the additive identity: $\vec{v} + \vec{0} = \vec{v}$.
Key Insight
The additive identity is unique: if both $e$ and $e'$ satisfy $a + e = a$ and $a + e' = a$ for all $a$, then $e = e + e' = e'$. This uniqueness proof is the same in every abelian group.
Definition
In abstract algebra, a group $(G, +)$ requires an identity element $0$ satisfying $a + 0 = 0 + a = a$ for all $a$ in $G$. A ring $(R, +, *)$ additionally requires $0 * a = 0$ for all $a$ (absorptive property of additive identity under multiplication). The additive identity is the kernel of the identity map and plays a central role in homomorphism theory.
Example
In the ring $M_2(\mathbb{R})$ of $2\times2$ matrices, the additive identity is the zero matrix $\begin{bmatrix}0&0\\0&0\end{bmatrix}$. Every matrix $A$ satisfies $A + 0 = A$. The zero matrix also absorbs multiplication: $0 \cdot A = A \cdot 0 = 0$.
Key Insight
The additive identity generates the trivial subgroup $\{0\}$ in any group. Every group homomorphism $f: G \to H$ must map $0_G$ to $0_H$ (the identity maps to the identity), a consequence of $f$ preserving the group operation.